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Session I.2 - Computational Number Theory

Wednesday, June 14, 17:00 ~ 17:30

Primitive points on elliptic curves

Peter Stevenhagen

Universiteit Leiden, Netherlands   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a point $P$ of infinite order on an elliptic curve $E$ defined over a number field $K$, one may ask, after Lang and Trotter, whether the set of primes $\frak p$ of $K$ for which the reduction of $P$ generates the point group over the residue class field of $\frak p$ possesses a density. Unlike the density for the set of primes of cyclic reduction of $E$, the heuristical density in this case has not been proven to be correct, not even under GRH.

We will focus on the vanishing of the heuristical density. This is a question that can be answered without assuming GRH. It has more subtleties than the density of the set of primes of cyclic reduction of $E$.

Joint work with Francesco Campagna (Leibniz Universitaet Hannover, Germany), Francesco Pappalardi (Roma 3, Italy) and Nathan Jones (University of Illinois at Chicago, USA).

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